Open-Circuit Fault-Tolerant Design of the Cascaded H-Bridge Rectiﬁer Incorporating Reactive Power Compensation

: This paper proposes an open-circuit fault-tolerant design for the cascaded H-Bridge rectiﬁer incorporating reactive power compensation. If one or two switching devices of the H-bridge modules are fault, the drive signals of the faulty H-bridge modules will be artiﬁcially redistributed into the bridgeless mode (including the boost bridgeless mode, the symmetric boost bridgeless mode, the totem-pole bridgeless mode and the symmetry totem-pole bridgeless mode) and cooperate with the normally operated H-bridge modules. In this case, the faulty cascaded H-bridge rectiﬁer is not only able to achieve active power transmission, but also can still provide part of reactive power compensation when injecting reactive power from the power grid. Nonetheless, the reactive power that it can supply will be limited, due to the unidirectional characteristics of the bridgeless mode for the faulty modules. Therefore, a method for calculating its adjustable power factor angle range is also presented, which provides the basis for the faulty modules switching to the bridgeless mode. Then, a control strategy of the cascaded H-bridge rectiﬁer incorporating reactive power compensation under the faulty condition and normal operation is presented. Finally, an experimental platform with a single-phase cascaded H-bridge rectiﬁer containing three cells is given to verify the proposed theories.


Introduction
In recent years, due to the access of nonlinear loads in the power system, the power quality control technology with power electronic converters as the core has attracted increasing attention. Increasing scholars and research teams believe that integrating reactive power compensation functionality with bidirectional rectifiers that exist in practice to construct multifunctional power electronic rectifiers in the distributed generation system is a more reasonable and cost-effective solution [1][2][3]. These multifunctional power rectifiers can not only realize their functions, but also complete the reactive power compensation for the grid. [4] has pointed out that the multifunctional power rectifiers are essential to employed in the smart grid and modular multicell converter to feature a two-way flow of electricity between production and consumption. [5,6] has pointed out that the Vienna rectifier and bridgeless rectifier and can be a multifunctional power rectifier with a power-quality added function. The advantage of the multifunctional power rectifier is that reactive power can be processed locally and distributed, thereby the loss of reactive power on power transmission line is reduced. In addition, realizing reactive power compensation requires almost no additional hardware costs [4][5][6]. Figure 1 shows the multifunctional cascaded H-bridge rectifier and its hybrid cascaded rectifier mode under open-circuit fault, wherein the power-switching devices are numbered as S k1 , S k2 , S k3 and S k4 (k = 1, . . . , m + n). u s is the input voltage. i s is the input current. L is the boost inductance. C is the capacitance in each module's DC-side. In the following paper, the value of DC capacitance in each module is equal. The output DC-side equivalent resistance for each module is R k (k = 1, . . . , m + n). u con is the AC-side voltage. There are eight states when one or two switches of the basic H-bridge module have an open-circuit fault. The basic H-bridge module can be artificially switched to the four bridgeless operation modes to overcome it, which includes the boost bridgeless, symmetric boost bridgeless, the totem-pole bridgeless and the symmetric totem-pole bridgeless operation modes. Table 1 shows the four-fault operation modes and the corresponding drive signals under these eight open-circuit fault states. The boost bridgeless operation mode and the symmetric boost bridgeless operation mode can adopt synchronous drive signals or complementary drive signals, while the totem-pole bridgeless operation mode and the symmetrical totem-pole bridgeless operation mode can only use the complementary drive signals to avoid the short-circuit fault caused by the direct connection of the upper and lower switching devices. Then the above faulty H-bridge module operated in the bridgeless operation mode can cooperate with the normally operated H-bridge module to achieve active power transmission and provide reactive power compensation. This faulty operation mode is defined as the hybrid cascaded rectifier mode. It should be noted that when S k1 and S k4 fail at the same time or S k2 and S k3 fail at the same time, the output level of each faulty module will be reduced. In this regard, cascaded H-bridge rectifier can operate normally through bypassing faulty modules.

Fault-Tolerant Design
Electronics 2020, 9, x FOR PEER REVIEW 3 of 15 Figure 1 shows the multifunctional cascaded H-bridge rectifier and its hybrid cascaded rectifier mode under open-circuit fault, wherein the power-switching devices are numbered as Sk1, Sk2, Sk3 and Sk4 (k = 1, …, m + n). us is the input voltage. is is the input current. L is the boost inductance. C is the capacitance in each module's DC-side. In the following paper, the value of DC capacitance in each module is equal. The output DC-side equivalent resistance for each module is Rk (k = 1,…, m + n). ucon is the AC-side voltage. There are eight states when one or two switches of the basic H-bridge module have an open-circuit fault. The basic H-bridge module can be artificially switched to the four bridgeless operation modes to overcome it, which includes the boost bridgeless, symmetric boost bridgeless, the totem-pole bridgeless and the symmetric totem-pole bridgeless operation modes. Table 1 shows the four-fault operation modes and the corresponding drive signals under these eight open-circuit fault states. The boost bridgeless operation mode and the symmetric boost bridgeless operation mode can adopt synchronous drive signals or complementary drive signals, while the totem-pole bridgeless operation mode and the symmetrical totem-pole bridgeless operation mode can only use the complementary drive signals to avoid the shortcircuit fault caused by the direct connection of the upper and lower switching devices. Then the above faulty H-bridge module operated in the bridgeless operation mode can cooperate with the normally operated H-bridge module to achieve active power transmission and provide reactive power compensation. This faulty operation mode is defined as the hybrid cascaded rectifier mode. It should be noted that when Sk1 and Sk4 fail at the same time or Sk2 and Sk3 fail at the same time, the output level of each faulty module will be reduced. In this regard, cascaded H-bridge rectifier can operate normally through bypassing faulty modules.

Analysis of Input Current Distortion for Hybrid Cascaded Rectifier Operation Mode
When the faulty cascaded H-bridge rectifier switches to the hybrid cascaded rectifier operation mode composed of m boost bridgeless modules and n H-bridge modules, which is shown in Figure 2, the input current distortion mechanism should be analyzed first. The output DC-side equivalent resistance for bridgeless modules is R i (i = 1, . . . m) and the output DC-side equivalent resistance for H-bridge modules is R j (j = m + 1, . . . m + n). u dci (i = 1, . . . m) and u dcj (j = m + 1, . . . m + n) are the DC-side voltages for bridgeless modules and H-bridge modules, respectively. u con_BR and u con_H are the AC-side voltage of bridgeless modules and AC-side voltage of H-bridge modules, respectively. It should be noted that the following analysis are also suitable for the m symmetry boost bridgeless modules, totem pole bridgeless modules or symmetry totem pole bridgeless modules and n H-bridge modules. Based on the assumption that the hybrid cascaded rectifier operates under the unity power factor, the steady-state AC-side phasor diagram is shown in Figure 3a. It is apparent that the power supply voltage Us is in phase with the command input current Is* and inductor voltage UL is orthogonal to Is*. According to the triangle law of vector addition, the AC-side reference voltage of the hybrid cascaded rectifier * con U is lagging Is* than γ. The AC-side reference voltage of bridgeless modules * _ con BR U is lagging Is* than α and the AC-side reference voltage of H-bridge modules Ucon_H* is lagging Is* than ψ. The unidirectional power transmission characteristics of the bridgeless modules will cause serious distortion in the period between the zero-crossing point of the input current is and the zero-crossing point of the ucon [23]. To make the bridgeless modules input current not distorted, the best way is to make * _ con BR U in phase with Is*as shown in Figure 3b [21,23]. In this case, bridgeless modules cannot provide the required reactive power of inductor L [22]. Moreover, it will be fully undertaken by the H-bridge modules. In the case of the AC-side reference voltage of the H-bridge module reaches its maximum value. However, the desired reactive power consumed by L that the Hbridge module should provide is not reached, the remaining reactive power consumed by L must be provided by the bridgeless module. In this case, the steady-state AC-side phasor diagram changes to Figure 3a. From the aforementioned analysis, input current will be distorted as shown in Figure 3c. Especially when injecting reactive power from the power grid, this distortion will greatly increase. Moreover, the adjustable power factor angle range that hybrid cascaded rectifier operation mode can provide will be limited. It is therefore important to calculate the adjustable power factor angle range to provide a basis for switching to the hybrid cascaded rectifier operation mode when the open-circuit fault happens. Based on the assumption that the hybrid cascaded rectifier operates under the unity power factor, the steady-state AC-side phasor diagram is shown in Figure 3a. It is apparent that the power supply voltage U s is in phase with the command input current I s * and inductor voltage U L is orthogonal to I s *. According to the triangle law of vector addition, the AC-side reference voltage of the hybrid cascaded rectifier U * con is lagging I s * than γ. The AC-side reference voltage of bridgeless modules U * con_BR is lagging I s * than α and the AC-side reference voltage of H-bridge modules U con_H * is lagging I s * than ψ. The unidirectional power transmission characteristics of the bridgeless modules will cause serious distortion in the period between the zero-crossing point of the input current i s and the zero-crossing point of the u con [23]. To make the bridgeless modules input current not distorted, the best way is to make U * con_BR in phase with I s *as shown in Figure 3b [21,23]. In this case, bridgeless modules cannot provide the required reactive power of inductor L [22]. Moreover, it will be fully undertaken by the H-bridge modules. In the case of the AC-side reference voltage of the H-bridge module reaches its maximum value. However, the desired reactive power consumed by L that the H-bridge module should provide is not reached, the remaining reactive power consumed by L must be provided by the bridgeless module. In this case, the steady-state AC-side phasor diagram changes to Figure 3a. From the aforementioned analysis, input current will be distorted as shown in Figure 3c. Especially when injecting reactive power from the power grid, this distortion will greatly increase. Moreover, the adjustable power factor angle range that hybrid cascaded rectifier operation mode can provide will be limited. It is therefore important to calculate the adjustable power factor angle range to provide a basis for switching to the hybrid cascaded rectifier operation mode when the open-circuit fault happens. Based on the assumption that the hybrid cascaded rectifier operates under the unity power factor, the steady-state AC-side phasor diagram is shown in Figure 3a. It is apparent that the power supply voltage Us is in phase with the command input current Is* and inductor voltage UL is orthogonal to Is*. According to the triangle law of vector addition, the AC-side reference voltage of the hybrid cascaded rectifier * con U is lagging Is* than γ. The AC-side reference voltage of bridgeless is lagging Is* than α and the AC-side reference voltage of H-bridge modules Ucon_H* is lagging Is* than ψ. The unidirectional power transmission characteristics of the bridgeless modules will cause serious distortion in the period between the zero-crossing point of the input current is and the zero-crossing point of the ucon [23]. To make the bridgeless modules input current not distorted, the best way is to make * _ con BR U in phase with Is*as shown in Figure 3b [21,23]. In this case, bridgeless modules cannot provide the required reactive power of inductor L [22]. Moreover, it will be fully undertaken by the H-bridge modules. In the case of the AC-side reference voltage of the H-bridge module reaches its maximum value. However, the desired reactive power consumed by L that the Hbridge module should provide is not reached, the remaining reactive power consumed by L must be provided by the bridgeless module. In this case, the steady-state AC-side phasor diagram changes to Figure 3a. From the aforementioned analysis, input current will be distorted as shown in Figure 3c. Especially when injecting reactive power from the power grid, this distortion will greatly increase. Moreover, the adjustable power factor angle range that hybrid cascaded rectifier operation mode can provide will be limited. It is therefore important to calculate the adjustable power factor angle range to provide a basis for switching to the hybrid cascaded rectifier operation mode when the open-circuit fault happens.

Analysis of Adjustable Power Factor Angle in the Hybrid Cascaded Rectifier Operation Mode
In the hybrid cascaded rectifier operation mode, * _ con BR U is always kept in phase with Is* to provide the active power required by DC-side loads, and the H-bridge modules provide the reactive power required by the AC-side and the active power required by the DC-side loads simultaneously. The AC-side phasor diagrams under lagging power factor and leading power factor are shown in Figure 4. The voltage across the input inductor UL is orthogonal to Is and the AC-side reference voltage * con U is lagging Us than φ. In the case of lagging power factor, φ < θ and φ ≥ θ may occur according to the different power, voltage level and power factor. The input current Is * is lagging grid voltage Us than θ. On the condition of leading power factor, the input current Is * is leading grid voltage Us than θ. The following relationships exist: Assuming the bridgeless modules can provide rms of the maximum AC-side reference voltage. the step-up ratio k is defined as: Considering that the DC-side voltages can be balanced in steady-state, the input current is derived by: Base on the analysis of Section 3. The AC-side voltage of the n H-bridge modules * _ con H U should not exceed the maximum total AC-side reference voltage, which can be expressed as:

Analysis of Adjustable Power Factor Angle in the Hybrid Cascaded Rectifier Operation Mode
In the hybrid cascaded rectifier operation mode, U * con_BR is always kept in phase with I s * to provide the active power required by DC-side loads, and the H-bridge modules provide the reactive power required by the AC-side and the active power required by the DC-side loads simultaneously. The AC-side phasor diagrams under lagging power factor and leading power factor are shown in Figure 4. The voltage across the input inductor U L is orthogonal to I s and the AC-side reference voltage U * con is lagging U s than ϕ. In the case of lagging power factor, ϕ < θ and ϕ ≥ θ may occur according to the different power, voltage level and power factor. The input current I s * is lagging grid voltage U s than θ.
On the condition of leading power factor, the input current I s * is leading grid voltage U s than θ. The following relationships exist: where U * coni (i = 1, ..., m) and U * conj (j = m + 1, ... m + n) are the AC-side reference voltage for each bridgeless module and H-bridge module, respectively.
Each module's DC-side output voltage is U dc . The maximum output voltages U conmax in rms of each cascaded module can be expressed as [24,26]: Assuming the bridgeless modules can provide rms of the maximum AC-side reference voltage. the step-up ratio k is defined as: Considering that the DC-side voltages can be balanced in steady-state, the input current is derived by: Base on the analysis of Section 3. The AC-side voltage of the n H-bridge modules U * con_H should not exceed the maximum total AC-side reference voltage, which can be expressed as:  In the case of φ < θ of lagging power factor operation, applying geometric relationship in ∆HPC yields: Substituting Equation (2), (3), (4), and (6) into (5) yields: On the condition of φ ≥ θ of lagging power factor operation, applying the geometric relationship in ∆HDC yields:  In the case of ϕ < θ of lagging power factor operation, applying geometric relationship in ∆HPC yields: Substituting Equation (2), (3), (4), and (6) into (5) yields: On the condition of ϕ ≥ θ of lagging power factor operation, applying the geometric relationship in ∆HDC yields: Due to the AC-side voltage of n H-bridge modules U * con_H satisfies (5), Substituting (3), (4), and (8) into (5) yields: In the case of leading power factor operation, hybrid cascaded rectifier's AC-side reference voltage u * con can be expressed as: u * con = U * con sin(ωt − ϕ) Electronics 2020, 9,1490 8 of 15 where the rms value of U * con and the corresponding lagging angle ϕ are derived by: Applying the law of cosines in ∆OHC, the AC-side reference voltage of the n H-bridge modules U * con_H can be expressed as: Substituting (3), (4), (11) and (12) into (5) yield: Equation (7), (9) and (13) shows that the adjustable lagging power factor angle and leading power factor angle θ is related to the step-up ratio k, inductance L and load resistance R i and R j . In most cases, the voltage drop on the boost inductor is usually small, so ϕ < θ is the common lagging power factor operation mode. By drawing the curve of the left Equation of (7) and (13) versus power factor angle θ and the curve of the right Equation of (7) and (13), respectively, the intersection point is the maximum lagging power factor angle θ lag_acmax and the maximum leading power factor angle θ lead_acmax that this hybrid cascaded rectifier operation mode can operate. In this case, the adjustable lagging power factor angle range and the adjustable leading power factor angle range are (0, θ lag_acmax ) and (θ lead_acmax , 0), respectively.
The adjustable power factor angle of the hybrid cascaded rectifier operation mode is not only affected by whether the AC-side voltage of the H-bridge modules can reach the maximum value which has been analyzed above, but also by the limit of the rated power S of the cascaded H-bridge rectifier. Then the maximum adjustable power factor angle can be expressed as: where P is the active power of the cascaded H-bridge rectifier. The adjustable lagging power factor angle range and the adjustable leading power factor angle range are (0, θ ra_max ) and (−θ ra_max , 0), respectively. Based on the above-mentioned two qualifications, the adjustable lagging power factor angle range is the interaction between the set of (0, θ ra_max ) and (0, θ lag_acmax ). Moreover, the adjustable leading power factor angle range is the interaction between the set of (−θ ra_max , 0) and (−θ lead_acmax , 0). For example, based on the parameters presented in Table 2, Figure 5 depicts the curves of AC-side voltage of the H-bridge modules U * con_H at different power factor angle θ with the various number of bridgeless modules m and H-bridge modules n under the same step-up ratio k or with various step-up ratio k under the same number of bridgeless modules m and H-bridge modules n in the lagging power factor operation and leading power factor operation. The solid line shows the left Equation of (7) and (13) versus θ. The dotted line depicts the right Equation of (7) and (13) versus θ. The intersection points (K, S, T, U, W, X, Y, Z) between the solid line and the dotted line is the maximum power factor angle that this hybrid cascaded rectifier operation mode can operate. Considering the limit of the rated power, the adjustable lagging power factor angle range and the adjustable leading power factor angle Electronics 2020, 9, 1490 9 of 15 range are (0, 58 • ) and (−58 • , 0 • ), respectively. The calculation results corresponding to Figure 5 and the final adjustable lagging power factor angle range and the final adjustable leading power factor angle range when considering the rated power restriction are shown in Table 3. Based on the above analysis, when the total number of cascaded modules is determined, the adjustable lagging power factor angle range and the adjustable leading power factor angle range that the hybrid cascaded rectifier operation mode can operate will increase with the increase of the number of H-bridge modules n or with the decrease of the step-up ratio k. The intersection points (K, S, T, U, W, X, Y, Z) between the solid line and the dotted line is the maximum power factor angle that this hybrid cascaded rectifier operation mode can operate. Considering the limit of the rated power, the adjustable lagging power factor angle range and the adjustable leading power factor angle range are (0, 58°) and (−58°, 0°), respectively. The calculation results corresponding to Figure 5 and the final adjustable lagging power factor angle range and the final adjustable leading power factor angle range when considering the rated power restriction are shown in Table 3. Based on the above analysis, when the total number of cascaded modules is determined, the adjustable lagging power factor angle range and the adjustable leading power factor angle range that the hybrid cascaded rectifier operation mode can operate will increase with the increase of the number of H-bridge modules n or with the decrease of the step-up ratio k.

Control Strategy
The multifunctional rectifier operates in the hybrid cascaded rectifier mode when one or two switches are faulty while it operates normally when there is no fault. The voltage and current double closed-loop control strategy can be used to achieve the control strategy. The active current reference i * sd in the current loop is derived by (15).
where K iP and K iI are the proportional coefficient and Integral coefficient, respectively. The reactive current reference i * sq depends on the desired reactive power. In this case, the power factor angle θ can also be defined as: By adopting the dq decoupling control, the active component and reactive component of the total AC-side reference voltage of the multifunctional cascaded rectifier, i.e., u * cond and u * conq , are obtained. In the case of the hybrid cascaded rectifier mode, u * con_BRd needs to be always in phase with i s to avoid input current zero-crossing distortion. The AC-side reference voltage of bridgeless modules u * con_BRd and u * con_BRq , as well as the AC-side reference voltage of the H-bridge modules u * con_Hd and u * con_Hq can be expressed as: If the system is unbalanced, the total input active power should be redistributed according to the different active power required by each bridgeless module and H-bridge module. The active-duty cycle should be modified by adding the voltage balance controller [26]. Then the new AC-side reference voltage of bridgeless module u * con_BRid , as well as the new AC-side reference voltage of H-bridge module u * con_H jd can be expressed as: where ∆d di (i = 1,...,m) and ∆d dj (j = m + 1,...,m + n) are the active regulation component and the reactive regulation component of duty cycles for the bridgeless modules and the H-bridge modules, respectively. By employing m + n inverse dq transformations, the final AC-side reference voltages, u * con_BRi and u * con_H j , can be obtained. In the case of normal operation and when the system is unbalanced, the AC-side reference voltage u * con_dk (k = 1, ..., m + n) for the cascaded H-bridge rectifier module is the sum of u * cond and ∆d di and the sum of u * cond and ∆d dj . By employing m + n single-phase inverse dq transformations, the final AC-side reference voltages u * con_Hk can be obtained. Then the unipolar modulation is adopted for the multifunctional cascaded H-bridge rectifier. This is because it possesses the merit of smaller switching loss and lower DC voltage ripple compared with bipolar modulation [27]. The control diagram of the multifunctional cascaded H-bridge rectifier under normal operation and the faulty condition is shown in Figure 6. can be obtained. Then the unipolar modulation is adopted for the multifunctional cascaded H-bridge rectifier. This is because it possesses the merit of smaller switching loss and lower DC voltage ripple compared with bipolar modulation [27]. The control diagram of the multifunctional cascaded H-bridge rectifier under normal operation and the faulty condition is shown in Figure 6.

Experiment Results
An experimental platform as shown in Figure 7 was established; its specifications are shown in Table 4. In the experimental tests, Modules 1 and 2 worked in bridgeless mode, while Module 3 works in H-bridge mode. REN-DSP28335 is selected as the core controller. First, considering the limit of rated power, the adjustable lagging and leading power factor angle range were 0 • -58 • and −58 • -0 • , respectively. Then it was essential to verify the calculated maximum power factor angle range when the cascaded H-bridge rectifier operated in the hybrid cascaded rectifier mode. Based on the analysis in Section 4, the adjustable lagging and leading power factor angle at which the hybrid cascaded rectifier operation mode could theoretically operate was 0 • -25 • and −27 • -0 • , respectively, which were in the range of the adjustable power factor angle ranges when considering the rated power. On the condition of the lagging power factor angle of the hybrid cascaded rectifier operation mode was 18 • , Figure 8a shows that the input current was sinusoidal and showing a satisfactory current quality. However, when the lagging power factor angle of the hybrid cascaded rectifier operation mode beyond the range of the adjustable power factor angle and was set as 35 • , Figure 8b shows that the input current becomes seriously distorted, which means this hybrid cascaded rectifier operation mode cannot provide such a large reactive power. In the case of the leading power factor angle of the hybrid cascaded rectifier operation mode was −24 • , Figure 9a shows that the input current was sinusoidal and showing a satisfactory current quality. However, when the leading power factor angle of the hybrid cascaded rectifier operation mode beyond the range of the adjustable power factor angle that the hybrid cascaded rectifier operation mode could operate and was set as -36 • , Figure 9b shows that the input current becomes distorted seriously, which means this rectifier cannot provide such a large reactive power. It is evident in Figure 8; Figure 9 that the input current was approximately sinusoidal within the range of the calculated adjustable power factor angle while the input current was seriously distorted beyond the calculated adjustable power factor angle in the hybrid cascaded rectifier operation mode. Figure 10 shows the maximum adjustable power factor comparison between the theoretical calculation values and experimental values under different step-up ratio k. It can be seen in Figure 10 that the experimental values were close to the theoretical calculation values. The waveforms of the DC-side voltage are shown in Figure 11. It is clear that under the voltage balance control presented in this paper, all the DC voltages were balanced, which proves the feasibility of the presented method.
was set as 35°, Figure 8b shows that the input current becomes seriously distorted, which means this hybrid cascaded rectifier operation mode cannot provide such a large reactive power. In the case of the leading power factor angle of the hybrid cascaded rectifier operation mode was −24°, Figure 9a shows that the input current was sinusoidal and showing a satisfactory current quality. However, when the leading power factor angle of the hybrid cascaded rectifier operation mode beyond the range of the adjustable power factor angle that the hybrid cascaded rectifier operation mode could operate and was set as -36°, Figure 9b shows that the input current becomes distorted seriously, which means this rectifier cannot provide such a large reactive power. It is evident in Figure 8; Figure 9 that the input current was approximately sinusoidal within the range of the calculated adjustable power factor angle while the input current was seriously distorted beyond the calculated adjustable power factor angle in the hybrid cascaded rectifier operation mode. Figure 10 shows the maximum adjustable power factor comparison between the theoretical calculation values and experimental values under different step-up ratio k. It can be seen in Figure 10 that the experimental values were close to the theoretical calculation values. The waveforms of the DC-side voltage are shown in Figure 11. It is clear that under the voltage balance control presented in this paper, all the DC voltages were balanced, which proves the feasibility of the presented method.

Conclusions
A new hybrid cascaded rectifier operation mode is proposed in this paper when the open-circuit fault happens in the cascaded H-bridge rectifier incorporating reactive power compensation. Then the adjustable power factor angle range for the hybrid cascaded rectifier operation mode can operate is calculated. It provides the basics about whether the cascaded H-bridge rectifier can switch to the hybrid cascaded rectifier mode when injecting reactive power from the power grid. Moreover, the control strategy for the cascaded H-bridge rectifier under the faulty operation and normal operation is presented. Finally, experiments verify the correctness of the calculated adjustable power factor angle range of the hybrid cascaded rectifier operation mode containing two bridgeless mode modules and one H-bridge mode module. Compared with the traditional faulty cascaded H-bridge rectifier without an adjustable power factor angle, the adjustable power factor angle range of the method presented in this paper is −27°-25°.

Conclusions
A new hybrid cascaded rectifier operation mode is proposed in this paper when the open-circuit fault happens in the cascaded H-bridge rectifier incorporating reactive power compensation. Then the adjustable power factor angle range for the hybrid cascaded rectifier operation mode can operate is calculated. It provides the basics about whether the cascaded H-bridge rectifier can switch to the hybrid cascaded rectifier mode when injecting reactive power from the power grid. Moreover, the control strategy for the cascaded H-bridge rectifier under the faulty operation and normal operation is presented. Finally, experiments verify the correctness of the calculated adjustable power factor angle range of the hybrid cascaded rectifier operation mode containing two bridgeless mode modules and one H-bridge mode module. Compared with the traditional faulty cascaded H-bridge rectifier without an adjustable power factor angle, the adjustable power factor angle range of the method presented in this paper is −27 • -25 • .
Author Contributions: T.C., H.C., W.C. and C.W. conceptualized the main idea of this project; T.C. proposed the methods and designed the work; T.C. conducted the experiments and analyzed the data; T.C. and W.C. designed the software; Z.Z. checked the results; T.C. wrote the whole paper; T.C., H.C. and C.W. reviewed and edited the paper. All authors have read and agreed to the published version of the manuscript.